3.3.1. Modeling of permanent magnet synchronous motor vector control.

The principle of space vector pulse width modulation (SVPWM) is based on vector equivalents. The magnitude and direction of the current vector can be indirectly controlled by the timing of the six switching elements of the inverter through the three-phase winding of the permanent magnet synchronous motor, so that the winding produces a constant amplitude circular magnetic field that rotates according to a given demand, thus dragging the permanent magnet to rotate. The voltage inverter circuit is shown in Figure 8. The simulation model of SVPWM control system for permanent magnet synchronous motor is shown in Figure 9.

#### 3.3.2. Modeling of cascade control system

The SVPWM control system for permanent magnet synchronous motor is based on the threephase current information and rotor position information fed back by the motor. AC motor is equivalent to a direct current motor by formula transformation to control the position and amplitude of the stator current.

The control system schematic is shown in Figure 10(a). The system includes a cascaded control structure with a P-position controller, a PI-velocity controller, and a PI-current controller. In the cascade control system, the servomotor feedback speed ω<sup>M</sup> and the work table feedback

position XT are calculated by the rotor position detected by the encoder on the servomotor. The input Pref is given by the CNC system according to the feed motion command, the input Pref and work table feedback position XT are compared, and the reference speed ωMref is given by the position controller. Then, the reference speed ωMref is compared with the feedback speed ωM, and the velocity controller gives the reference current iqref for the q-axis and the reference current idref ¼ 0 for the d-axis of the stator. The three-phase current of the servomotor is detected and converted into id and iq in d � q coordinate system through the Clark and Park

Electromechanical Co-Simulation for Ball Screw Feed Drive System

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Figure 10. Schematic of ball screw feed drive system electromechanical co-simulation.

Figure 9. Simulation model of SVPWM control system.

Figure 8. Circuit of voltage bridge inverter.

Figure 9. Simulation model of SVPWM control system.

Te ¼ 1:5p ψ<sup>f</sup> iq þ Ld � Lq

In order to model the servo control of the ball screw feed system, the modeling of the threeloop cascade control architecture of the vector control and servo control system of the permanent magnet synchronous servomotor is studied, which is commonly used in the ball screw

The principle of space vector pulse width modulation (SVPWM) is based on vector equivalents. The magnitude and direction of the current vector can be indirectly controlled by the timing of the six switching elements of the inverter through the three-phase winding of the permanent magnet synchronous motor, so that the winding produces a constant amplitude circular magnetic field that rotates according to a given demand, thus dragging the permanent magnet to rotate. The voltage inverter circuit is shown in Figure 8. The simulation model of SVPWM control system for permanent magnet synchronous motor is shown in Figure 9.

The SVPWM control system for permanent magnet synchronous motor is based on the threephase current information and rotor position information fed back by the motor. AC motor is equivalent to a direct current motor by formula transformation to control the position and

The control system schematic is shown in Figure 10(a). The system includes a cascaded control structure with a P-position controller, a PI-velocity controller, and a PI-current controller. In the cascade control system, the servomotor feedback speed ω<sup>M</sup> and the work table feedback

3.3. Servo control modeling of ball screw feed system

3.3.2. Modeling of cascade control system

amplitude of the stator current.

Figure 8. Circuit of voltage bridge inverter.

3.3.1. Modeling of permanent magnet synchronous motor vector control.

feed system.

50 New Trends in Industrial Automation

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(29)

Figure 10. Schematic of ball screw feed drive system electromechanical co-simulation.

position XT are calculated by the rotor position detected by the encoder on the servomotor. The input Pref is given by the CNC system according to the feed motion command, the input Pref and work table feedback position XT are compared, and the reference speed ωMref is given by the position controller. Then, the reference speed ωMref is compared with the feedback speed ωM, and the velocity controller gives the reference current iqref for the q-axis and the reference current idref ¼ 0 for the d-axis of the stator. The three-phase current of the servomotor is detected and converted into id and iq in d � q coordinate system through the Clark and Park

Figure 11. Electromechanical co-simulation model of half-closed ball screw feed system.

transformation. idref and iqref are compared with the feedback id and iq, respectively, and the current controller calculates the given voltages Ud and Uq of the d and q axes; then, they are converted into U<sup>α</sup> and U<sup>β</sup> in the α β coordinate system by Park inverse transformation. Finally, the SVPWM module generates six-phase PWM to drive the three-phase inverter. The inverter outputs ABC three-phase voltage to servomotor stator, which generates rotating magnetic field and produces magnetic torque on the servomotor rotor. This magnetic torque is the output torque TM of the servomotor and drives the rotor to rotate under the dynamic relations of ball screw feed system.

servo system of Shenyang Machine Group, which use a semi-closed-loop cascade control structure. The specifications of the test bench are listed in Table 1, which are either obtained from the manufacturers' catalogs, approximated from prior knowledge, or calculated from computer-aided design (CAD). According to the modeling method described in Chapter 2, the lumped mass model of this ball screw feed system test bench was built up. The equivalent parameters of the lumped mass model were calculated by using the specifications in Table 1,

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Taking the servomotor torque as input and the axial acceleration of work table as output, the frequency response characteristics of the lumped parameter model of the test bench are analyzed. The bode diagram is shown in Figure 13, and simulation result shows that the work table has four-order natural frequencies, which are 26.2, 76.7, 247, and 633 Hz. Further study shows that 76.7 Hz is the main axial vibration frequency of the work table, 26.2 Hz is the main axial vibration frequency of the base, and 247 and 633 Hz are the rotational vibration frequencies.

Parameter of the component Value Parameter of the component Value Work table mass MT (kg) <sup>206</sup> Rotary inertia of coupling <sup>J</sup><sup>C</sup> (kg <sup>m</sup>2) <sup>1</sup>:<sup>09</sup> <sup>10</sup><sup>4</sup> Base mass MB (kg) <sup>3820</sup> Rotary inertia of motor JM (kg <sup>m</sup>2) <sup>6</sup>:<sup>75</sup> <sup>10</sup><sup>3</sup> Coupling mass Mc (kg) 1.18 Torsional rigidity of coupling kc (N=m) <sup>1</sup>:<sup>4</sup> 103 Motor rotor mass Mm (kg) 10.9 Screw bearing rigidity kb (N=m) <sup>1</sup> 108 Screw pitch length <sup>h</sup> (m) <sup>1</sup>:<sup>6</sup> <sup>10</sup><sup>2</sup> Nut reference rigidity <sup>K</sup> (N=m) <sup>6</sup>:<sup>12</sup> <sup>10</sup><sup>8</sup> Screw diameter ds (m) <sup>2</sup>:<sup>5</sup> <sup>10</sup><sup>2</sup> Nut basic dynamic load Ca (N) 37.4 Screw length ls (m) 1 Ball screw length at table position ln (m) 0.35

and the other calculated lumped parameters are listed in Table 2.

Figure 12. Single-axis ball screw feed drive test bench.

Table 1. Specifications of the test bench.
